How do we find Integrating factor for a General Diff equation

[himanshu121]

For eg is there a way to find IF for
$$pydx +qxdy +x^my^n(rydx+sxdy)=0$$

 

[HallsofIvy]

It can be proved that every first order d.e. has an integrating factor but there is no general way to do it.

 

[saltydog]

I have some questions:

(1) What are p,q,r, and s?

Constants or functions?

(2) The proof that you mention Hall, is it difficult to follow? For example, is it more involved then the general existence and uniqueness proofs for a first order ODE?

(3) If a solution can be shown to exist, then does this somehow guarantee the existence of some integrating factor? For example, the above equation can be written as:

$$\frac{dy}{dx}=-\frac{py+x^mry^{n+1}}{qx+sx^{m+1}ry^{n+1}}$$


If the RHS of this equation and it's partial with respect to y are bounded in some neighborhood of the x-y plane, then a solution can be shown to exists in that neighborhood. Is that connected with showing an integrating factor exists as well?

(4) To Himanshu:

How close to the above equation can you get with another equation that looks like it and still be able to determine an integrating factor?

 

[mathelord]

collect like terms and express the equation in terms of mdx+ndy,since you need integrating factors,it will be e^(integral of (1/n)(dm/dy -dn/dx).you still need to get this confirmed.and ask for more help from hall,saltydog,matt grime and especially hurkyl.they are the best

 

[saltydog]

collect like terms and express the equation in terms of mdx+ndy,since you need integrating factors,it will be e^(integral of (1/n)(dm/dy -dn/dx).you still need to get this confirmed.and ask for more help from hall,saltydog,matt grime and especially hurkyl.they are the best

Dude, you obviously must have just skipped your fingers randomly across the keyboard and they just happen to randomly spell "saltydog" cus' I'm not in the same category as those guys. But nice to be around them though.

 

[lurflurf]

For eg is there a way to find IF for
$$pydx +qxdy +x^my^n(rydx+sxdy)=0$$

I do not know exactly what you are asking. There is no way to find an integrating factor for a general differential equation. That is there is no general method to find the integrating factor of any differential equation. An integrating factor for your example differential equation can be found by a non-general method.
$$pydx +qxdy +x^my^n(rydx+sxdy)=0$$
First we can observe this.
$$x^{1-p}y^{1-q}{d}(x^py^q)=pydx+qxdy$$
Then we can write the equation as
$$xy(x^{-p}y^{-q}{d}(x^py^q)+x^my^nx^{-r}y^{-s}{d}(x^ry^s))=0$$
so that is is easy to see that there will be an integrating factor of the form
$$\frac{(x^py^q)^a}{xy}$$
to find a we note
$$(x^py^q)^a=x^my^n(x^ry^s)^b$$
so we solve
pa=m+rb
and
qa=n+sb
The equation times the integrating factor can then be easily integrated.
Note: There is a potential problem if sp=rq

 

[lurflurf]

collect like terms and express the equation in terms of mdx+ndy,since you need integrating factors,it will be e^(integral of (1/n)(dm/dy -dn/dx).you still need to get this confirmed.and ask for more help from hall,saltydog,matt grime and especially hurkyl.they are the best

This will not work because and integrating factor that depends on x and y is needed. That ony works when there exist an integrating factor whose mixed second partial is 0. That if if there is an integrating factor of the form u(x) or u(y). Here we need an integrating factor of the form (x^py^q)^a.